Why won't eyes move with the rest of my mesh?
In short, make sure that other bones aren't also affecting the mesh where the eyes are sitting.

More specifically, you should parent the eyes to whatever you have set as the head's bone as well. Ensure that other bones aren't affecting the mesh of the head. Visually, you would want the area around the eyes paint weighted red, and the eyes as well. When reviewing the other vertex groups, the area around the eyes should be blue, with 0 weights set to that location. This will ensure nothing else is affecting that area.